Counting And Cardinality

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Article Title: K, Counting and Cardinality; K-5, Operations, and Algebraic Thinking

Summary

The article describes counting and cardinality, operations and algebraic thinking as a way

of understanding using numbers. It analyses several progressions that arise from understanding the

names of the number and the count sequence:

K.CC.1.

The progression in operations basically

involves addition, subtraction, multiplication and division. The first operations to be taught are

addition and subtraction whereby subtraction is comprehended as a reverse in whatever is done in

addition. Likewise, students learn the meaning of multiplication and division where division is a

reverse in the actions carried out in multiplication. Comprehending all these arithmetic properties

helps students build and improve their mathematical knowledge and are therefore able to solve

various mathematical problems in real life (Barchers 24).

How the Article relates to what I Have Learned over the Ten Weeks

Over the ten weeks, I have learned the progression in Operations as well as Algebraic

Thinking, dealing with the fundamental operations. The article proposes that most standards that

fall under the Operations and Algebraic heading mostly involve whole numbers. However,

progression is generally significant as it explains concepts, representations, and properties which

extend to other systems of numbers, to algebra and measures.

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Furthermore, progressions use whole numbers, decimals, variables, and even fractions. In

Operations and Algebraic Thinking, students first learn addition and subtraction. While addition

means increasing something, subtraction attempts to reverse the actions of addition. Likewise, the

meaning of multiplication and division differs since the student understands division as a reverse

action of multiplication.

The article also gives out the origin of the various progressions through the knowledge of

number names and the counting sequence. It helps to understand the relationship between numbers

and quantities thus giving out a very clear connection between counting and cardinality. It also

points out the concept of comparison of various numbers. The student is able to identify whether

the objects in a certain group are greater than, less than or equal to the number of objects in another

group (Barchers 26).

How the Article Relates to How My Ideas about Math, Teaching, and Learning Have

Changed over The Period?

The article relates much to how my ideas about Math, teaching, and learning have

significantly changed over the period. For instance, it proposes that students learn the properties

of arithmetic over time, which include commutativity and associativity of addition and

multiplication, as well as distributivism of multiplication over addition. Initially, learners

intuitively understand the properties and use the gained understanding in solving mathematical

and real-world problems (The Common Core Standards Writing Team 2).

The article has also changed my idea of teaching Math to students. It proposes that various

progressions originate in understanding names of numbers and the count sequence. It is, therefore,

necessary to teach learners first how to say the counting words and then how to count out objects.

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They learn the counting words up to a certain number, before using the numbers to count objects.

The article has also changed my idea of Math since it explains that learners establish meanings for

addition and subtraction as they continue facing problem situations from kindergarten, and later

advance in grade 1 as they face more challenging problems (The Common Core Standards Writing

Team 3).

What I Have Learned About Student Thinking About the Mathematical Topic at Hand from

Reading about this Section

I have learned that students identify the small group’s cardinalities without counting the

objects, through perceptual subtilizing which later advances into conceptual subtilizing. They

recognize that a collection of objects comprises of two sub-collections ((Teacher Created Materials

45). They, therefore, quickly combine their cardinalities to look for the collection’s cardinality.

For instance, they see a set as two subsets of cardinality 2 and say “four.” Furthermore, students

compare numbers

K.CC.6, K.CC.7

through identification of the group with more than, fewer than or an

equal amount as the other. They understand that matching or counting can show a different

outcome even if one group seems to have more objects than the other (The Common Core

Standards Writing Team 5).

I have also learned about the various method that students can use to solve single digit

addition and subtraction problems. They can use the direct modelling method by counting all the

objects and then subtracting. They can also use the counting on method in order to get the total or

find an addend. Lastly, they can convert the addend to a simpler one which is equivalent to the

former problem (Teacher Created Materials 56).

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Questions That I Have After the Reading

After reading this article, I developed two key questions. For instance, how can we

determine that a certain whole number ranging from one to hundred is a multiple of a certain one-

digit number? The author does not give a clear explanation of this. He argues that one has to

interpret prior understanding of division in terms of the language of factors and multiples which

does not explain it all. (The Common Core Standards Writing Team 30).

Furthermore, how can students develop an understanding of operations and algebraic

thinking? The author explains that the pervasive use of mathematical practices by students in every

grade allows them to understand operations and algebraic thinking. However, he does not explain

clearly how it happens. He, therefore, needs to provide more information on the same so as to

clear any doubts and questions that might arise.

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Works Cited

The Common Core Standards Writing Team "Progressions for the Common Core State

Standards for Mathematical (Draft)." Teaching to the Math Common Core State

Standards, 2011, pp. 2-31.

Barchers, Suzanne. Counting and Cardinality: More or Less Practice. Teacher Created

Materials, 2014.

Teacher Created Materials, Incorporated. Counting and Cardinality, Geometry for Kindergarten

Set. 2018.

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